Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

Abstract

We are concerned with singular limits of stiff relaxation and dominant diffusion for general 2 × 2 nonlinear systems of conservation laws, that is, the relaxation time y tends to zero faster than the diffusion parameter e, y=o(e), e Q 0. We establish the following general framework: If there exists an a priori L . bound that is uniformly with respect to e for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other type of limits. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L . bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in L p to the equilibrium solutions of these systems is established, provided that the relaxation time y tends to zero faster than e.

Nonstrictly hyperbolic systems with stiff relaxation terms

“…In this case, a general convergence framework for the relaxation-viscosity (v τ,ε , u τ,ε ) of the Cauchy problem (1.4), (1.2) was established in [8], where system (1.1) could be elliptic. Some results for 3 × 3 systems are given in [9,11].…”